Best Known (32, 32+30, s)-Nets in Base 64
(32, 32+30, 513)-Net over F64 — Constructive and digital
Digital (32, 62, 513)-net over F64, using
- t-expansion [i] based on digital (28, 62, 513)-net over F64, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- the Hermitian function field over F64 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
(32, 32+30, 515)-Net in Base 64 — Constructive
(32, 62, 515)-net in base 64, using
- (u, u+v)-construction [i] based on
- (5, 20, 257)-net in base 64, using
- base change [i] based on digital (0, 15, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- base change [i] based on digital (0, 15, 257)-net over F256, using
- (12, 42, 258)-net in base 64, using
- 2 times m-reduction [i] based on (12, 44, 258)-net in base 64, using
- base change [i] based on digital (1, 33, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- base change [i] based on digital (1, 33, 258)-net over F256, using
- 2 times m-reduction [i] based on (12, 44, 258)-net in base 64, using
- (5, 20, 257)-net in base 64, using
(32, 32+30, 1777)-Net over F64 — Digital
Digital (32, 62, 1777)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6462, 1777, F64, 2, 30) (dual of [(1777, 2), 3492, 31]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6462, 2053, F64, 2, 30) (dual of [(2053, 2), 4044, 31]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6462, 4106, F64, 30) (dual of [4106, 4044, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(6462, 4107, F64, 30) (dual of [4107, 4045, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(25) [i] based on
- linear OA(6459, 4096, F64, 30) (dual of [4096, 4037, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(6451, 4096, F64, 26) (dual of [4096, 4045, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(643, 11, F64, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,64) or 11-cap in PG(2,64)), using
- discarding factors / shortening the dual code based on linear OA(643, 64, F64, 3) (dual of [64, 61, 4]-code or 64-arc in PG(2,64) or 64-cap in PG(2,64)), using
- Reed–Solomon code RS(61,64) [i]
- discarding factors / shortening the dual code based on linear OA(643, 64, F64, 3) (dual of [64, 61, 4]-code or 64-arc in PG(2,64) or 64-cap in PG(2,64)), using
- construction X applied to Ce(29) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(6462, 4107, F64, 30) (dual of [4107, 4045, 31]-code), using
- OOA 2-folding [i] based on linear OA(6462, 4106, F64, 30) (dual of [4106, 4044, 31]-code), using
- discarding factors / shortening the dual code based on linear OOA(6462, 2053, F64, 2, 30) (dual of [(2053, 2), 4044, 31]-NRT-code), using
(32, 32+30, 2978303)-Net in Base 64 — Upper bound on s
There is no (32, 62, 2978304)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 9619 662472 048643 711988 507866 399216 116800 973003 778759 216183 109164 797137 103269 295195 201693 385494 476012 204375 718849 > 6462 [i]